New Spain is a vast area, yet I’ve had difficulty finding scholarship that discusses how the Spanish secured their position militarily. What military forces were available to the viceroy? Was there a sizable presence of Spanish regular forces, or would military forces have been primarily locally-raised units? How did Spanish colonial authorities defend their territory against other European powers and Native groups?
Can't give a comprehensive answer (not being a military historian and due to the expansive scope of the question), but one interesting example I am familiar with would be the series of Spanish forts in northern Florida. In particular, the fort at St. Augustine was actually protected by another fort just to the north of the settlement known as Fort Mose. In contrast to the fort inside the city which was staffed by Spanish soldiers, Fort Mose was primarily staffed by escaped African slaves who fled the British colonies to the north (particularly the Carolinas and Georgia). The Spanish actively encouraged these maroons by offering their freedom so long as they swore fealty to the Spanish crown and became Catholic. In return, they both weakened the British colonial economy (however marginally) and helped increase the number of soldiers they could keep on the borders of La Florida. This was particularly helpful in Florida were Spanish settlements were sparse and the number of Spanish soldiers was generally equally sparse throughout the 18th and 17th centuries.
The first issue is that there were two kinds of targets.
The first were rich ports which could be sacked. Havana, Panama, and other important ports were sacked through the colonial period. Some were even sacked several times. These were mostly limited to raids by pirates and privateers, and most of these events occurred in the 16th and 17th centuries.
The second kind of attack was an attempt by a foreign power to actively seize and hold a territory. These mostly targeted the Antilles which were both largely undefended and more easily defensible than a mainland territory so long as the nation in question held naval superiority over Spain. A naval power could capture an island, say Barbados, put a few marines in a makeshift fort and set a few warships to defend it until the claim was begrudgingly accepted by Spain.
In rarer cases naval expeditions would go against areas which were either on the mainland (as in the case of the British invasions of Buenos Aires in the early 19th century) or at least against one of the larger Antilles (such as when Havana was invaded and successfully occupied). These were rarer because it was more difficult to hold more densely populated territories with three or four thousand British marines. British naval superiority in regards to Spain would, thus, mean less.
The case of Florida, cited above, is a bit different since the peninsula was also largely unpopulated, save for a few fort towns (Edit: Obviously when I say "unpopulated" I am referring to the Spanish population, since many Native Americans, who were fiercely independent, also lived there). Even so, in all three cases sparse Spanish troops from the peninsula were supplemented by creole aid (criollo being the term for Spanish subjects born in the Americas). I couldn´t speak to their importance in Florida, but in the case of Cuba, when Havana was invaded in 1762 by English forces, creoles took upon themselves a constant guerrilla war despite the formal surrender of the official Spanish captain general. This is why the British were able to capture Havana but not extend their holdings very far inland. Even today the figure of "Pepe Antonio" leading the resistance against English troops in Havana from the nearby settlement of Guanabacoa is a popular image. Creoles were also extremely important in the defense of Buenos Aires, with a local militia successfully beating hardened British marines whose leaders expected them to be received with open arms. A local militia, it should be said, who were created in part in response to the important defeats of the 18th century, such as the siege and capture of Havana by England.
Important coastal towns were raided less and less as time went on because the Spanish Bourbon dynasty established a series of forts at the most important links in the commercial chain linking Spain with its American colonies. Havana got El Castillo del Morro. Along the coast of New Spain they fortified San Juan de Ullua, which fate would have it became the last Spanish stronghold on the mainland during the Hispanic American wars of independence. In Colombia (then New Granada), Cartagena was fortified. This was part of a huge empire wide reform in that went hand in hand with the economic reforms the Bourbons enacted.
All in all, however, there were rarely any real dangers that a heavily populated part of the mainland empire would fall to foreign marines. Even as late as the early 19th century, invasion forces of more than 20,000 were rare. Both Rafael de Riego´s force (which was supposed to retake Spanish America but instead started a liberal revolution in Spain) and the British invasion of New Orleans during the War of 1812 were less than 30,000. These were enough to maybe take a city, but couldn´t hold large amounts of the interior without local support.
Finally, in regards to black military support, these could also be found in Buenos Aires and Havana under the official title of "El Regimento de Pardos y Morenos" or "the regiment of blacks and browns" (blacks and mixed race, respectively). The latter was abolished in Cuba only in the 19th century, but the exact moment escapes me. I have a pretty good idea, but I won´t speculate because of the subreddit rules. (EDIT: Apologies, they were the Batallones de Pardos y Morenos, so battalions and not regiments)
Bibliography:
--Tulio Halperin Donghi: The History of Contemporary Latin America, Duke University Press, 1993. The title says "contemporary", but it talks about the Bourbon reforms and military changes as well as comparing Spain´s response throughout its empire.
-Arthur P. Whitaker: The United States and the Independence of Latin America, 1800-1830, W. W. Norton & Company, Inc., LaVergne, 2011. Whitaker talks about British attempts to take Spanish territory and the Spanish response in the first chapters.
-Jacobo de la Pezuela: Historia de la Isla de Cuba, Carlos Beilly-Bailliere, Madrid, 1878. Volumes 2-4. As one of the "Ayacucho Generals" (generals defeated during the independence wars), Spanish historian Jacobo de la Pezuela is obsessed with military issues affecting Cuban history and touches on defenses and military structures throughout his immense multi-volume work.
-Manuel Moreno Fraginals: Cuba/España España/Cuba: Historia Común, Editorial Crítica, Barcelona, 1995. Moreno does a great job discussing the changing structural relationship between Cuba and Spain, including the concrete affects of the Bourbon reforms.
Adding to /u/RioAbajo/ 's answer from the perspective of a broader regional scope, the short answer is, they didn't really. At the same time, most of the areas claimed as colonial holdings were not yet of great interest to other powers. This gives the appearance of Spanish hegemony over vast areas both on land and at sea, which was not true. Similarly, French explorers claimed large tracts of land in the middle of North America, which was neither defensible, nor was attractive to other European powers at the time. Here's a simplistic map showing the situation in 1776 when the Thirteen Colonies revolted against Colonial England. The French were present in French north America in certain population centers, but never did they have control over all of the territories they claimed.
In today's US Pacific Coast, and former dominions of New Spain such as today's US states of New Mexico, Arizona, Texas, etc. most of these holdings consisted of missions run by Catholic missionaries. "Spanish" settlers were never large in numbers. There were not even many conquistadors at all. Most of the activities were done by native Americans with nominal loyalty to the viceroyalty of New Spain. And as mentioned by /u/RioAbajo/ , in some cases even by runaway slaves. The expansive claim over land that isn't secured is what eventually led to the Mexican-American war, although that is a separate subject. This post by /u/anthropology_nerd/ is a very good exposition on Spanish activity on the northern frontier of their domain.
Spain's interests were concentrated to several centers such as Havana, Vera Cruz, Puebla, San Domingo, and on the west coast Acapulco as a port for the Manila Galleons. The Florida fort that was mentioned was considered especially important as it protected Havana, which became the muster point for Spanish ships for their convoy back to Seville.
In the early part of the conquest of Mexico, the Spanish benefited from their alliance with the Tlaxcala, you can read through this post here. Tlaxcalan soldiers were used in the expansion of Spanish dominion. Quoting from that post,
In 1591, Tlaxcala settlers were granted further rights, so they could be used to peacefully colonize other tribes, by having them settle among them, thus allowing them to be introduced to a peaceful, agricultural, Christian lifestyle. These settlers gained some special rights, such as being exempted from Spanish taxes, getting control of various Chichimeca resources, property rights of the land seized from the Chichimeca and being allowed to ride on horseback and use guns. These Tlaxcalan settlers settled among other places along the Rio Grande.
At that time, meaning the 16th and 17th centuries, Spain's exclusive access was due to their control of sea lanes and lack of bases for other powers to operate from. Here's a good post on English failures in the 17th century, when Cromwell attempted to invade Santo Domingo.
It was not until the Bourbon Reform that the colonies start to organize their militia strongly, or more precisely that they were allowed to have a system in which they maintained a militia, but this happened only in the 1740s-1750s. Prior to this period, there was significant jockeying for control between the Habsburg kings in Spain and the local powers in the Americas. This led to inter-colony trade restrictions and limits on local powers including militia power restrictions.
